Predicting multilayer film's residual stress from its monolayers

Materials and Design 110 (2016) 858–864

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Predicting multilayer film's residual stress from its monolayers C.Q. Guo, Z.L. Pei ⁎, D. Fan, R.D. Liu, J. Gong, C. Sun ⁎ Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Residual stress in a multilayer film equals the weighted average of its monolayers' stresses. • Relative errors in predicting multilayers' stresses are in the range of 0.2% to 10.7% in this paper. • Alternating multilayers' stresses gradually approach a constant value as its monolayer number increases. • Si-DLC interfaces can either rise or lower DLC films' residual stresses.

a r t i c l e

i n f o

Article history: Received 24 May 2016 Received in revised form 12 August 2016 Accepted 15 August 2016 Available online 17 August 2016 Keywords: Residual stress Multilayer film Diamond-like carbon CrN/DLC multilayer Cathodic vacuum arc

a b s t r a c t Multilayer film's residual stress was deduced from Stoney formula. A simple stress formula, which means that multilayer residual stress can be given by the weighted average of each monolayer's residual stress, was proposed and verified through experiments on gradient diamond-like carbon (DLC) and CrN/DLC multilayers prepared by cathodic vacuum arc technology. Typical stress formulas for alternating multilayers were also investigated on corresponding DLC multilayers. Multilayer samples, together with monolayers existed in multilayers, were prepared and studied. Surface profilometry and film stress tester were used to measure films' thicknesses and residual stresses, respectively. Cross-sectional morphologies of multilayers were observed by scanning electron microscope. Results showed that the proposed stress formula was correct and could provide useful instructions on multilayer design. The formula's accuracy of predicting multilayer's residual stress through its monolayers was also investigated. In the present paper, relative errors of theoretical values were in the range of 0.2% to 10.7%, which had a strong relationship with the substrate–film interfaces. In addition, as to alternating multilayer film, its residual stress is a constant value as the number of monolayers is even; while this number is odd, multilayers' residual stress gets close to the constant value gradually and monotonously. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Residual stresses in films have been of interest to scientists for a long time, especially for multilayers [1–6]. Proper residual stress is good to raise films' toughness and adhesion to substrates [7,8], while ⁎ Corresponding authors. E-mail addresses: [email protected] (Z.L. Pei), [email protected] (C. Sun).

http://dx.doi.org/10.1016/j.matdes.2016.08.053 0264-1275/© 2016 Elsevier Ltd. All rights reserved.

excessively high stress may lead to film failure [9–11]. Therefore, lots of researchers tried to take efforts to predict residual stress to design multilayers with high performance. Numbers of methods for predicting residual stresses in multilayer films or structures have been proposed. In the research of Hsueh [12], an exact closed-form solution was formulated to predict thermal stress in elastic multilayer systems. Strain distribution in the system was decomposed into a uniform strain component and a bending strain

C.Q. Guo et al. / Materials and Design 110 (2016) 858–864

component. Zhang et al. [13] proposed an analytical model based on force and moment balances to predict thermal residual stress distributions in multilayered coating systems. A closed-form solution of thermal stress was obtained which is independence of the number of coating layers. They also used a similar model to predict distribution and magnitude of thermal residual stress in multilayer coatings with graded properties and compositions [14]. Systematical analysis of effects of the gradient exponent, elastic modulus of ceramic component, number of coating layers and substrate's properties on thermal residual stress was conducted on ZrO2Y2O3/NiCrAlY functionally and compositionally graded thermal barrier coatings. In another study, Zhang et al. [15] put forward a numerical model to predict the thermally induced residual stresses in the multilayer coating on a substrate with cylindrical geometry. This model is based on that the axial forces in the longitudinal direction and the interfacial pressures in the radial direction, which were derived from differential thermal contraction between the adjacent layers, could be determined by the continuity conditions at the interfaces using layer-by-layer procedure. In addition, finite element simulation has also been used to predict residual stresses in thermal barrier coatings [16–18]. Though all these methods can help people understand residual stress distribution or magnitude within multilayers, they either can't be applied to films with high intrinsic stresses or can't provide specific value when some parameters involved in the model are hard to obtain. In the present study, a concise and practical method based on that Young's modulus and Poisson ratio of the multilayer-substrate system are approximately equal to that of substrate was put forward. A weighted average formula of multilayer's residual stress derived from Stoney formula was presented and then verified by gradient amorphous (DLC) as well as composite (CrN/DLC) multilayers. What's more, alternating multilayer's stress formulas derived from the former weighted average formula were also displayed and investigated through corresponding multilayer DLC films. By analyzing relative errors of theoretical values, the formulas' feasibility of predicting multilayer's residual stress was discussed in detail.

σ ab ¼

2. Theory and experimental details

σ abc ¼

2.1. Theory

σ¼

Et 2s 1 6ð1−υÞ t f




 1 1 ; − R R0

ð1Þ

where tf is the thickness of film, R0 and R correspond to radii of curvature of substrate measured before and after film deposition [19]. As for certain substrates, Young's modulus (E), Poisson's ratio (υ) and thickness (ts) are all invariable values. So the part Et2s /[6 (1 − υ)] in Eq. (1) can be replaced by a constant: K. Hence, the modified Stoney formula could be rewritten as K σ¼ tf

Fig. 1. Schematic diagram of a gradient multilayer film.

(1) t f ¼ ∑ ti≪ts. i

(2) Young's modulus and Poisson's ratio of the system composed of film and substrate are approximately equal to that of substrate. (3) No cracks or delamination occur in the film–substrate system. (4) Deformation of substrate is in the range of elastic deformation. Table 1 shows residual stresses of multilayers as well as substrate's radii of curvature with corresponding multilayers. Substrate's radius of curvature changes from R0 to RA after layer A is deposited. Then, following equations can be obtained, according to Eq. (2). σa ¼

K ta

 1 1 : − R R0

ð2Þ

Under harsh conditions, films made up by one single layer sometimes can't meet the needs any more. Fig. 1 illustrates a common design of a gradient multilayer film consisting of four kinds of monolayers: A, B, C and D. Here, ti and σi represent thickness and residual stress of a certain monolayer, respectively. Several assumptions are made for later derivation.

1 1 − RA R 0

 ð3Þ




1 1 − RB R0

K ðt a þ t b þ t c Þ

 ð4Þ




1 1 − RC R0

K ðt a þ t b þ t c þ t d Þ

 ð5Þ




1 1 − RD R0

 ð6Þ

The first and second assumptions suggest that when layer (i + 1) is deposited, the original substrate together with layers from 1 to i acts as new substrate for layer (i +1). Therefore, residual stresses of monolayers B, C and D can be expressed as σb ¼

K tb

σc ¼

K tc




 1 1 ; − R B RA




1 1 − RC RB

ð7Þ

 ð8Þ

and σd ¼







K ðt a þ t b Þ

σ abcd ¼

When using Stoney formula to calculate residual stress in a film, it is considered that thickness of the film is much less than that of substrate [4]. The modified Stoney formula can be written as follows:

859

K td




 1 1 : − RD RC

ð9Þ

Table 1 Residual stresses of multilayers and corresponding radii of curvature. Multilayers

AB

ABC

ABCD

Residual stress Radius of curvature

σab RB

σabc RC

σabcd RD

860

C.Q. Guo et al. / Materials and Design 110 (2016) 858–864

According to Eqs. (3) to (9), residual stresses (σab, σabc and σabcd) of multilayers can be represented by residual stresses (σa, σb, σc and σd) and thicknesses (ta, tb, tc and td) of its monolayers: σ ab ¼ σ abc

σ a ta þ σ b tb ; ta þ tb

ð10Þ

σ a ta þ σ b tb þ σ c tc ¼ ; ta þ tb þ tc

σ abcd ¼

σ a ta þ σ b tb þ σ c tc þ σ d td : ta þ tb þ tc þ td

ð11Þ ð12Þ

As a result, it can be easily seen from Eqs. (10), (11) and (12), n X

σ 1;2; :::n ¼

;

ð13Þ

ti

i¼1

where σ1 , 2 , ... n is residual stress of a multilayer film in which the number of monolayers is n (n ≥ 1, and n takes an integer value). That is, a multilayer film's residual stress depends upon all its monolayers' residual stresses and thicknesses. Alternating multilayer film consisting of two kinds of monolayers (A and D) with different residual stresses (σa ≠ σd) illustrated by Fig. 2 is often designed to relieve residual stress. Here, m (m ≥ 1, and m takes an integer value) represents the number of bilayers in an alternating multilayer film. Thus, residual stress of the multilayer film can be represented by σ1 , 2 , … 2m (the number of monolayers is even) or σ1,2, …(2m+1) (the number of monolayers is odd). According to Eq. (13), σ 1;2;…2m ¼

mσ a t a þ mσ d t d σ a t a þ σ d t d ¼ ; mt a þ mt d ta þ td

ð14Þ

and σ 1;2;…ð2mþ1Þ ¼

  lim σ 1;2;…2m −σ 1;2;…ð2mþ1Þ ¼ 0:

m→þ∞

ð17Þ

Consequently, as to alternating multilayer film, residual stress in odd-numbered multilayers changes monotonously with increased layer number, and gradually approach the stress value of the even-numbered multilayers. 2.2. Experimental details

σ i ti

i¼1 n X

Obviously, the value of (σ1 , 2 , … 2m − σ1 , 2 , … (2m + 1)) drops monotonously with the increasing of the bilayer number m, which means that extent of variation of residual stress in alternating multilayer film decreases as film thickness rises. As a result, it becomes equal to zero when m tends towards infinitude:

ðm þ 1Þσ a t a þ mσ d t d : ðm þ 1Þt a þ mt d

ð15Þ

From Eqs. (14) and (15), clear conclusions can be drawn. If the number of monolayers is even, residual stress of the multilayer is a constant, which is only related to its two monolayers' residual stresses and thicknesses and has no connection with the bilayer number m. Otherwise, it will vary with m. Hence, σ 1;2;…2m −σ 1;2;…ð2mþ1Þ ¼

ðσ d −σ a Þt a t d : ðt a þ t d Þ½ðm þ 1Þt a þ mt d 

Fig. 2. Schematic diagram of an alternating multilayer film.

ð16Þ

Two kinds of multilayer films (gradient multilayer film—sample Ι and sample II, alternating multilayer film—sample ΙIΙ) with the structure exhibited in Figs. 1 and 2 were deposited on P (100) Si wafers. Samples I and III were multilayer DLC films prepared by a filtered cathodic vacuum arc (FCVA) ion-plating apparatus with a 90° bend plasma duct fitted between the cathodic arc source and the coating chamber. While in sample II consisting of three CrN layers (layers A⁎, B⁎ and C⁎) and one DLC layer (layer A), CrN layers were prepared by a direct cathodic vacuum arc (DCVA) source. As to sample ΙIΙ, four bilayers composed of layer-A and layer-D stack on Si substrate. The coating equipment had been described in detail in previous research [20]. Before film deposition, the chamber was evacuated to 4.0 × 10−3 Pa. When depositing DLC films high-purity Ar (99.999%) was chosen as working gas to sustain arc discharge with a working pressure of about 0.1 Pa, while it was high-purity N2 (99.999%) with a working pressure of about 0.7 Pa for CrN films. DLC monolayers A, B, C and D with different residual stresses were deposited under different substrate bias voltage (− 600 V, − 400 V, − 150 V and − 100 V, respectively). Similarly, CrN monolayers A⁎, B⁎ and C⁎ were prepared also by varying substrate bias voltage (−200 V, −400 V and − 600 V). Duty ratio and repetition frequency were kept invariable (30% and 50 kHz, respectively) throughout deposition procedure. Different thicknesses of layers were controlled by varying deposition time. Si wafers with the size of 25 × 5 × 0.4 mm3 were ultrasonically cleaned in acetone and ethanol for about 15 min respectively, then surveyed by film stress tester (FST 150, SPI, China) for radius of curvature (R0). Each time before closing the chamber gate, three Si wafers were put into the equipment for different purpose. One was for multilayer deposition. Another was used to measure residual stress of a monolayer on Si substrate. The third one with partly covered by aluminum foil was to gain thickness of a monolayer. When the first monolayer was prepared, all the samples were taken out of the chamber. Then the two Si wafers without aluminum foil were surveyed again by FST 150 for radius of curvature immediately. Subsequently, another two new clear Si substrates, of which one was partly covered by aluminum foil, were put into the chamber together with the sample which was used for multilayer deposition. After that, next monolayer could be prepared. This process was repeated until the multilayer film deposition was completed. Finally, three multilayer samples (samples Ι, ΙΙ and III), as well as sixteen monolayer samples, were prepared. Throughout the deposition process, no plasma etching was performed to avoid affecting film stress and thickness. Surface profilometry (Alpha-step IQ, KLA Tencor, USA) with a resolution of 0.0328 nm and a repeatability of 0.1% was used to investigate the thicknesses of films. Stylus force, scan length and scan speed were 29.7 mg, 1000 μm and 50 μm/s, respectively. Step profiles of three points were collected for each film. Residual stresses of all films or layers were also calculated by FST 150 based on Stoney equation. Biaxial modulus of Si wafers is 180.5 GPa [21,22]. The results had a margin of error of plus

C.Q. Guo et al. / Materials and Design 110 (2016) 858–864

or minus 2.0%. Cross sectional morphology and structure of multilayer films were observed by field emission scanning electron microscopy (SEM, Inspect F50, FEI, USA). 3. Results and discussion 3.1. Gradient multilayer film 3.1.1. Verification and application of the method for amorphous multilayer Fig. 3a illustrates the step profiles of gradient DLC multilayer (sample I) and monolayers on Si substrates measured by surface profilometry. The left part was bare substrate, and the right part was covered with film. Thicknesses of films were gained by comparing the different heights between the two parts: 570.6 nm (tabcd), 230.3 nm (ta), 175.0 nm (tb), 78.7 nm (tc) and 86.6 nm (td). From the cross-sectional morphology of sample Ι showed in Fig. 3b, it is easy to see that four monolayers stacked on Si substrate. No pores or cracks were observed both inside the monolayers and at the interfaces. It also confirmed the values of thicknesses. Gradient multilayer DLC films' residual stresses (σ1, 2, ...n) as well as that of monolayers (σi) in sample Ι were obtained by FST 150 based on Stoney formula through corresponding radius of curvature (R0, RA, RB, RC, RD) and thickness (ta, tb, tc, td, tabcd). Values of residual stresses are displayed in Table 2. According to Eq. (13), value of residual stress in multilayer (σ1,2, ...n) could also be calculated through its monolayers' thicknesses (ti, measured by surface profilometry) and residual stresses (σi, presented in Table 2). As expected, the two series of results—gaining through FST 150 and calculating with Eq. (13)—have the same values when their accuracy is three significant figures after decimal point. This suggests that the assumption made in Section 2.1 is rational and the derivation process of Eq. 13 is correct and logical. However, when multilayer film is completed, it is inconvenient and meaningless that gaining multilayer's residual stress through its monolayers' thicknesses and stresses instead of calculating directly with Stoney formula. In most cases, thicknesses and residual stresses of monolayers are well studied before multilayer film is prepared. Therefore, residual stress in multilayer film can be predicted in advance with Eqs. (13), (14) or (15), where significance of the formulas lies. Application of Eq. (13) in calculating multilayers' residual stresses was investigated in detail through residual stresses in monolayers (A, B, C and D) directly deposited on Si wafers (σ⁎i ). Relative errors (δ) of theoretical values (σ1,2, ...n⁎) were obtained with the following formula: δ =(|σ1,2, ...n⁎ − σ1,2, ...n |/σ1 ,2, ...n) × 100%, which were shown in Table 2. For a better understanding of the difference between measured residual stresses and theoretical values of multilayers, results shown in Table 2 are illustrated in Fig. 4. The theoretical values were a little lower than measured values but still very close. That means, it is feasible to predict multilayer film's residual stress by Eq. (13).

861

Table 2 Residual stresses of monolayers (σi, σ⁎i) and gradient DLC multilayers (σ1,2, ...n, σ1,2, ...n⁎). Monolayer

A

B

C

D

σi (−GPa) σ⁎i (−GPa)

1.555 1.555 – – – –

1.959 1.863 AB 1.729 1.688 2.4%

3.455 2.937 ABC 2.010 1.891 5.9%

4.322 4.141 ABCD 2.361 2.233 5.4%

Multilayer σ1 ,2, ...n (−GPa) σ1 ,2, ...n⁎ (−GPa) δ

A closer look at Fig. 4 shows that residual stresses of monolayers on Si substrates (σb⁎, σ⁎c and σd⁎) are much lower than that of monolayers in sample Ι (σb, σc and σd), which leads to previous relative errors showed in Table 2. Considering the same deposition parameters for a certain monolayer, it is the difference between Si\\C interface and C\\C interface which leads to the residual stresses of unequal value. Probably, the different thermal expansion coefficients and atomic distances between DLC and Si substrate as well as mutual diffusions between carbon atoms and silicon atoms [23] contribute to the different residual stresses of monolayers deposited simultaneously. Moreover, though compressive stress of layer D is up to 4.322 GPa, that of gradient multilayer (ABCD) is 2.361 GPa. That is, introducing intermediate layers is an effective method to control film's residual stress. 3.1.2. Verification and application of the method for composite multilayer Sample II containing three CrN layers (layer A⁎, B⁎ and C⁎) and one DLC layer (layer A) was prepared to check whether Eq. (13) is supported by crystalline and composite multilayers. Step profiles of the four monolayers measured by surface profilometry were presented in Fig. 5a, in which thicknesses could be easily gained: 174.1 nm (ta⁎), 107.7 nm (tb⁎), 116.4 nm (tc⁎) and 126.3 nm (ta). Cross-sectional morphology of sample II was observed by SEM and shown in Fig. 5b. The interface between DLC and CrN is clear compared with the blurry interfaces between CrN monolayers. Thicknesses of layer A and sample II presented in Fig. 5b consist with the results shown in Fig. 5a. Residual stresses of composite multilayers (σ1,2 , ...n) and monolayers in sample II (σi) presented in Table 3 were calculated by FST 150 based on Stoney formula from corresponding thickness and radius of curvature. According to Eq. (13), values of σ1, 2, ...n could also be obtained by residual stresses (σi) and thicknesses (ti) of its monolayers. Multilayers A⁎B⁎ and A⁎B⁎C⁎ are crystalline films, while multilayer A⁎B⁎C⁎A (sample II) is a kind of composite film. The equality of the two series of results with three significant figures after decimal point suggests that Eq. (13) is applicable to both crystalline and amorphous multilayers. Feasibility of predicting composite multilayers' residual stresses (σ1 , 2 , ... n⁎) from monolayers' thicknesses (ti) and residual stresses (σ⁎i ) deposited directly on Si wafers was also investigated. Relative errors (δ) of theoretical values (σ1 ,2, ...n⁎) together with σ1,2, ...n⁎ and σ⁎i are presented in Table 3. The maximum value of relative error is 7.8% which

Fig. 3. (a) Step profiles of gradient multilayer, monolayers A, B, C and D. (b) Cross-sectional morphology of sample Ι.

862

C.Q. Guo et al. / Materials and Design 110 (2016) 858–864 Table 3 Residual stresses of monolayers (σi, σ⁎i) and composite multilayers (σ1,2, ...n, σ1,2, ...n⁎). Monolayer

A*

B*

C*

A

σi (−GPa) σ⁎i (−GPa)

2.735 2.735 – – – –

3.03 3.044 A⁎B ⁎ 2.848 2.853 0.2%

2.421 2.809 A⁎B⁎C⁎ 2.723 2.84 4.3%

1.001 1.379 A⁎B⁎C⁎A 2.308 2.488 7.8%

Multilayer σ1 ,2, ...n (−GPa) σ1 ,2, ...n⁎ (−GPa) δ

sample ΙIΙ were raised compared with that of sample I, correctness of Eq. (13) is still well verified.

Fig. 4. Evolution curve of residual stress versus thickness for gradient multilayer film.

originates from the different residual stresses in monolayers whether they were deposited directly on Si wafers or existed in multilayer films. These small relative errors showed that predicting residual stress in composite multilayers according to Eq. (13) could be realized.

3.2. Alternating multilayer film 3.2.1. Verification of the method To differentiate each layer in sample ΙIΙ, the monolayers were named as A1, D1, A2, D2, A3, D3, A4 and D4 according to deposition order. Multilayers of ADAD, ADADA, ADADAD, ADADADA and ADADADAD can be abbreviated as 2(AD), 2(AD)A, 3(AD), 3(AD)A and 4(AD). Step profiles of alternating multilayer 4(AD) and monolayers (layer A2 and D2) are displayed in Fig. 6a. Thickness of 4(AD) is 693.2 nm. Fig. 6b presents cross-sectional morphology of sample ΙIΙ, in which the boundary from layer A to layer D is much clearer than that from layer D to A. Thicknesses (ti) and residual stresses (σi) of monolayers in sample ΙIΙ as well as alternating multilayers' residual stresses (σ1 , 2 , ... n) are displayed in Table 4, in which residual stresses were obtained by FST 150 based on Stoney formula. Similar to the results of gradient multilayer, alternating multilayers' residual stresses calculated with Eq. (13) through monolayers' thicknesses (ti) and residual stresses (σi) are the same with values (σ1,2, ...n) in Table 4 when they have three significant figures after decimal point. Though both thickness and layer number of

3.2.2. Application of the method Feasibility of application of Eqs. (14) and (15) in calculating alternating multilayers' residual stress was also studied. Residual stresses of monolayers on Si substrates (σ⁎i ) and alternating multilayers (σ1 ,2 , ...n⁎, which was calculated with Eqs. (14) and (15)) are illustrated in Table 4. To reduce the influence of experimental error, ta and σa in Eqs. (14) and (15) refer to average thickness (86.8 nm) and residual stress (− 1.369 GPa) of monolayers (A) on Si substrates, while td (86.5 nm) and σd (−4.437 GPa) correspond to that of monolayers (D). Relative errors (δ) of σ1,2, ...n⁎ were also calculated and then presented in Table 4. In order to get a deeper insight into the evolution of alternating multilayers' residual stress versus thickness, values in Table 4 are showed in Fig. 7. The difference between theoretical values and measured values is very small. In addition, compressive stresses of even-numbered multilayers are very close to the theoretical value (2.9 GPa). While compressive stresses of multilayers ADA, 2(AD)A and 3(AD)A rise with the increased layer number, approaching this theoretical value (2.9 GPa) gradually. This result is well consistent with the derivation in Section 2.1. Lots of researchers think that stress relaxation by multilayer structuring is due to the view plastic deformation is easy to occur for the monolayer with lower stress [24–26]. Eqs. (14) and (15) provide a theoretical support for this explanation. Another interesting phenomenon is that as for monolayers A2, A3 and A4, compressive stresses of monolayers in sample ΙIΙ are always lower than that of monolayers on Si substrates. While for monolayers D1, D2, D3 and D4, the opposite is the case. That is, Si-DLC interface can either rise or lower residual stress in DLC film. Before this paper, lots of researchers have investigated residual stresses in alternating multilayer films. Table 5 presents the measured values (σ1,2, ...2m) reported in literatures and the theoretical values calculated with Eq. (14) (σ1 , 2 , ... 2m⁎). It's obvious to see, though bilayer number (from 2 to 30), thickness ratio (from 1:1 to 1:5) and monolayers' stress (from −0.8 to −13.7 GPa) varied a lot, theoretical values

Fig. 5. (a) Step profiles of monolayers A⁎, B⁎, C⁎ and A. (b) Cross-sectional morphology of sample ΙI.

C.Q. Guo et al. / Materials and Design 110 (2016) 858–864

863

Fig. 6. (a) Step profiles of multilayer 4(AD), monolayers A2 and D2. (b) Cross-sectional morphology of sample III.

were still very close to the values measured in these studies. All relative errors are lower than 10%. In addition, refs. [24,28,29] also plotted evolution curves of residual stress versus film thickness for alternating multilayer films, which displayed similar tendency to that showed in Fig. 7. With the increase of monolayer number, residual stress in alternating multilayer film got close to a constant gradually. This provides more evidences that the formulas proposed in Section 2.1 not only present new methods to calculate residual stress in multilayer film (which equals the weighted average of its monolayers' residual stresses) but also help researchers understand multilayer's residual stress more deeply (how it changes as layer number increases). For a multilayer film, Young's modulus n X

Ef ¼

Ei t i

i

n X

;

ð18Þ

ti

i

where Ei is a monolayer's Young's modulus [30]. According to the critical thickness formulas [31] t max ¼

2E f γ f σ2

ðtensile‐stressed filmÞ

ð19Þ

4. Conclusions A new method for assessment of residual stress in multilayer film derived from Stoney formula has been put forward and verified by gradient as well as alternating multilayer films, which provides a feasibility of predicting multilayer film's residual stress without doing real experiments and helps operators optimize multilayer's design. The proposed multilayer stress formulas are on the base that Young's modulus and Poisson ratio of the system consisting of substrate and film are approximately equal to that of substrate. A multilayer film's residual stress can be given by the weighted average of residual stresses in its monolayers. As to alternating multilayer films, if the number of monolayers is even, its residual stress is a constant value; while the number of monolayers is odd, its residual stress rises or declines monotonously with the increasing layer number, getting close to the former constant value gradually. That means, alternating multilayer film's residual stress does not grow infinitely with film deposition, but tends to a constant. However, it has restrictions when predicting a multilayer film's residual stress through monolayers on Si wafers or other kinds of substrates according to multilayer stress formulas. Relative errors of theoretical values mainly come from the difference between monolayers deposited directly on substrates and corresponding monolayers existing in multilayer film. In other words, the smaller this difference is, the more accurate the weighted average formula of multilayer stress is.

and

t max ¼

  E f γ f þ γs σ2

ðcompressive‐stressed filmÞ;

ð20Þ

where γf is surface free energy of the film, γs corresponds to surface free energy of the substrate, the rough maximum thickness (tmax) of the film without cracking or spalling can be calculated. This presents useful information to experimenters on how to deposit multilayer films with proper thicknesses and improve their research plan. Table 4 Thicknesses (ti) and residual stresses (σi, σ⁎i) of monolayers as well as alternating multilayers' residual stresses (σ1,2, ...n, σ1,2, ...n⁎). Monolayer

A1

D1

A2

D2

A3

D3

A4

D4

ti (nm) σi (−GPa) σ⁎i (−GPa) Multilayer σ1 ,2, ...n (−GPa) σ1 ,2, ...n⁎ (−GPa) δ

89.8 1.019 1.019 – –

83.5 4.775 4.473 AD 2.828

88.3 0.844 1.395 ADA 2.159

89.4 4.677 4.567 2(AD) 2.800

81.3 0.927 1.683 2(AD)A 2.448

93.0 4.399 4.147 3(AD) 2.793

87.7 0.651 1.377 3(AD)A 2.487

80.2 4.947 4.562 4(AD) 2.772

–

2.900 2.389 2.900

2.594

2.900

2.681

2.900

–

2.6%

6.0%

3.8%

7.8%

4.6%

10.7% 3.6%

Fig. 7. Evolution curve of residual stress versus thickness for alternating multilayer film. Squares linked by a solid line represent measured residual stresses of multilayers; circles linked by a dash-dotted line represent theoretical values calculated from Eqs. (14) and (15). Horizontal solid lines represent residual stresses of monolayers in sample III; horizontal dash-dotted lines represent residual stresses of monolayers on Si substrates.

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Table 5 Measured values of residual stresses in some previous studies (σ1,2, ...2m) and theoretical values calculated according to Eq. (14) (σ1,2, ...2m⁎). Refs.

Bilayer number (m)

Alternating multilayer

Thickness ratio (ta:td)

Residual stress ratio (σa:σd)

σ1,2 , ...2m (−GPa)

σ1, 2, ...2m⁎ (−GPa)

Relative error (δ)

[27]

30 15 2 3 3 3 8

(soft/hard) DLC

1:1

(soft/hard) DLC (soft/hard) DLC (TiC/DLC) (soft/hard) DLC (soft/hard) DLC

1:5 1:1 1:1 1:2 5:23

1.6:8.2 5.5:8.2 0.8:4.5 1.0:7.7 4.0:13.7 1.9:6.2 2.0:5.6

4.5 7.0 3.6 4.5 8.5 4.8 5.2

4.9 6.9 3.9 4.4 8.9 4.8 5.0

8.9% 1.4% 8.8% 2.2% 4.7% 0 3.8%

[25] [24] [26] [28] [29]

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